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\hfill \textbf{March 22, 2012}

\vspace{0.5 cm} \centerline{ \textbf{\large{Homework 4\hspace{1 cm}
Math 6303}}}

\vspace {1 cm}
\begin{enumerate}
\item Characterize those finite abelian
groups that are not the direct sum of
non-trivial subgroups.

\item Let $m_{\tau}(x)$ be the minimal
polynomial of the linear map $\tau$ on the
finite dimensional vector space $\mbf{V}$.
Show that $\tau$ is bijective iff
$m_{\tau}(0)\ne0$. Conclude that for a
bijective map $\tau$, the map $\tau^{-1}$ is
a polynomial in $\tau$, i.e.,
$\tau^{-1}=p(\tau)$ for some $p(x)\in
\mbf{F}[x]$.
\item Let $(\mbf{V},\tau)$ be as
before. Show that $\mbf{V}=\mbf{V}_0\oplus
\mbf{V_1}$ where $\mbf{V}_0$ and $\mbf{V}_1$
are invariant subspaces of $\mbf{V}$ and
where $\tau\upharpoonright \mbf{V}_0$ is
nilpotent, i.e., $\tau^{k}=0$ for some $k$
and where $\tau\upharpoonright \mbf{V}_1$ is
bijective.
\item Find for any polynomial
$p(x)\in \mbf{F}[x]$ some $(\mbf{V},\tau)$
such that the minimal polynomial of $\tau$ is
$p(x)$.
\item List all linear maps on $\mbb{R}^{2}$
and on $\mbb{R}^{3}$ according to their
similarity types.
\item For a linear map
$\tau$ on the finite dimensional vector space $\mbf{V}$ a
subspace $\mbf{U}$ is called \textit{indecomposable} if
$\mbf{U}$ is \textbf{not} the direct sume of two non-zero
invariant subspaces. Use a dimension argument in order to show
that $\mbf{V}$ is a direct sum of indecomposable subspaces.
Prove: $\mbf{U}$ is indecomposable if and only if $\mbf{U}$ is
cyclic and the minimal polynomial is a prime power.
\item Assume that $\tau$ is
diagonalizable. Is it true that every
invariant subspace is diagonalizable? Prove
your answer.

 \end{enumerate}
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\item Let $\mbf{G}$ and $\mbf{H}$ be finite
abelian groups such that for every integer $n$, $\mbf{G}$ and
$\mbf{H}$ have the same number of elements of order $n$. Prove that
$\mbf{G}\iso \mbf{H}$.
